Cohomology of Linear Cycle Sets when the adjoint group is finite abelian

This paper analyzes the second cohomology group of a linear cycle set with coefficients in an abelian group I, for linear cycle sets with commutative adjoint operation, focusing on the finite abelian case. It aims to classify extensions of such structures through cohomological methods. Techniques are developed to systematically construct explicitly 2-cocycles. Finally, some illustrative examples are explored to validate the theoretical framework.